ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 60S- 



that is to say, the radius of gyration relatively to Ov is the length of the 

 perpendicular A P dropped on this line from the extremity of its conju- 

 gate diameter : hence this tangent coincides with K K'. 



Jnst as the property of the ellipse of stress is only a special case 

 for a plane area of the more general property for solids of the ellipsoid of 

 stress, so the ellipse of inertia is only the special case for a plane area of 

 the more general property for solid bodies of the ellipsoid of inertia or 

 ' momental ' ellipsoid, the ' central ' or ' equiraomental ' ellipsoid corre- 

 sponding for a solid body to the central ellipse for a plane area. 



If we suppose segments of opposite sign, it is possible to obtain an 

 hyperboloid of inertia and central hyperboloid ; but this, as Routh 

 remarks,^ does not occur in practice, since the moment of inertia is 

 essentially positive, being by definition the sum of a number of squares, 

 and it is clear that every radius vector must be real. Hence the quadric 

 is always an ellipsoid. 



The properties already referred to for the central ellipse have their 

 counterpart for the central ellipsoid ; thus at any point of a space through 

 which segments can be drawn there are always three principal axes at 

 right angles to one another. 



The above author goes on to deal with what corresponds to the 

 products of an infinite number of segments, explaining the method given 

 by Bresse.^ 



Let 7] be the abscissa of the centroid of the given section. 

 Then 



7? £?«% = 



>/dxdy. 



Consider a cylinder with a base corresponding to the given area and 

 its generatrices normal to the plane of the area, and having a boundary 

 plane passing through Ox and inclined at 45° to the given plane. Let 

 r]y be the projection of the centroid of the cylinder on the given plane. 



Volume of the elementary cylinder having as base dxdy is 



Y=:7^dxdy. 

 The moment about 



0.v^^t)-dxdy. 

 But 



r,Mydxdy=\y/dxdy. 



Multiplying the corresponding sides of the last two equations, and sup- 

 pressing the common factor I ydxdy, we have 



r],]A\dxdy={\y-dxdy ; 



hence 



vni 



y^dxdy 

 \\dxdy 



' Migid Dynamics, 3rd edit., p. 16. 



- Cours de Resistance de Matcriartx prof esse a VEcole drs Fonts et Chavssees. 



